How do you find the derivative of #y=1+x^-1+x^-2+x^-3#?

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Answer 1 · 2018-05-09T01:57:37

The first derivative is #-(x^2+2x+3)/x^4#.

#d/dx[x^n]=nx^(n-1)#

Here's the derivative worked out (I color-coded some parts so that they would be easier to follow):

#color(white)=d/dx[color(red)1+color(orange)(x^-1)+color(green)(x^-2)+color(blue)(x^-3)]#

#=color(red)(d/dx[1])+color(orange)(d/dx[x^-1])+color(green)(d/dx[x^-2])+color(blue)(d/dx[x^-3])#

#=color(red)0+color(orange)((-1)(x^(-1-1)))+color(green)((-2)(x^(-2-1)))+color(blue)((-3)(x^(-3-1)))#

#=color(orange)(-x^-2)color(green)-color(green)(2x^-3)color(blue)-color(blue)(3x^-4)#

#=-(color(orange)(x^-2)+color(green)(2x^-3)+color(blue)(3x^-4))#

#=-(color(orange)(1/x^2)+color(green)(2/x^3)+color(blue)(3/x^4))#

#=-(color(orange)(x^2/x^4)+color(green)((2x)/x^4)+color(blue)(3/x^4))#

#=-(color(orange)(x^2)+color(green)(2x)+color(blue)3)/x^4#

That's the derivative. Hope this helped!